Module MATH3471: Geometry of Mathematical Physics III

Department: Mathematical Sciences

MATH3471: Geometry of Mathematical Physics III

Prerequisites

• Analysis in Many Variables II (MATH2031) AND (Mathematical Physics II (MATH2071) OR Theoretical Physics 2 (PHYS2631)).

Corequisites

• None.

Excluded Combination of Modules

• None.

Aims

• To introduce students to algebraic and geometric structures that arise in modern mathematical physics.
• To explore the role of symmetries in physical problems and how they are formulated in mathematical terms, focussing on examples from classical field theory such as electromagnetism.

Content

• Lie groups, Lie algebras, and representations.
• Representations of U(1), SU(2) and the Lorentz group, including spinors.
• Variational principle for fields and symmetries.
• Constructing variational principles invariant under symmetries.
• Charged particle in electromagnetic field and gauge symmetry.
• Variational principle for abelian gauge symmetry.
• Non-abelian gauge theory, their coupling to charged fields and variational principle.
• Some examples of topologically non-trivial field configurations: abelian Higgs model, Wu-Yang monopole,'t Hooft Polyakov monopole, Bogomolnyi monopoles,

Learning Outcomes

• Upon completion of the course, students will:
• have a conceptual understanding of Lie groups and Lie algebras.
• be familiar with how representation theory is applied in fundamental physics, with the Lorentz group and spinors as a specific example.
• understand the formulation of gauge theories using variational principles.

• the students will be able to solve problems in theoretical physics by using methods from group theory and representation theory.

• to formulate and analyse field theories based on symmetry principles.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Problem classes show how problems of varying difficulty can be approached and solved.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and a high level of rigour as well as feedback for the students and the lecturer on the students’ progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Eight assignments to be submitted.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University